"If I start right now, it is going to take me at LEAST an hour to clean this half alone, then it will take another half hour to clean half of the remainder, and 15 mins to clean half of THAT remainder... since I will always have half left, I will never be done!"

Do you agree with Sayber? Will Sayber be stuck with a vacuum in his hand forever?

Tune in next week...

Watch This

Guidance

Let’s return to the situation in the introduction: Poor Sayber is stuck cleaning his room. He cleans half of the room in 60mins. Then he cleans half of what is left, 30 more minutes, half again for 15 more. If he keeps cleaning half of the remaining area, how will he ever finish the room?

We know that the pieces have to add up to some finite time period (no matter what it feels like, Sayber CAN get the room clean), but how is it possible for the sum of an infinite number of terms to be a finite number?

To find the sum of an infinite number of terms, we should consider some partial sums. Three partial sums, relatively early in the series, could be: \begin{align*}S_2 = 90, S_3 = 105,\end{align*}S2=90,S3=105, and \begin{align*}S_6 = 118.125\end{align*}S6=118.125 or \begin{align*}118 \frac{1}{8}\end{align*}11818

As n approaches infinity, the value of Sn seems to approach 120 minutes. In terms of the actual sums, what is happening is this: as n increases, the nth term gets smaller and smaller, and so the nth term contributes less and less to the value of Sn . We say that the series converges, and we can write this with a limit:

Therefore, no matter how long the process continues, Sayber will not spend more than 2hrs cleaning the room. Of course, it may SEEM like a lot more!

We can do the same analysis for the general case of a geometric series, as long as the terms are getting smaller and smaller. This means that the common ratio must be a number between -1 and 1: |r| < 1.

Therefore, we can find the sum of an infinite geometric series using the formula \begin{align*}S = \frac{a_1} {1 - r}\end{align*}S=a11−r.

When an infinite sum has a finite value, we say the sum converges. Otherwise, the sum diverges. A sum converges only when the terms get closer to 0 after each step, but that alone is not a sufficient criterion for convergence. For example, the sum \begin{align*}\sum_{n = 1}^\infty \frac{1} {n} = 1 + \frac{1} {2} + \frac{1} {3} + \frac{1} {4} + ....\end{align*} does not converge.

b) The series 3 + -6 + 12 + -24 + ... does not converge, as the common ratio is -2.

Remember that the idea of an infinite sum was introduced in the context of a realistic situation, albeit a paradoxical one. We can in fact use infinite geometric series to model other realistic situations. Here we will look at another example: the total vertical distance traveled by a bouncing ball.

Example C

A bouncing ball

A ball is dropped from a height of 20 feet. Each time it bounces, it reaches 50% of its previous height. What is the total vertical distance the ball travels?

Solution

We can think of the total distance as the distance the ball travels down + the distance the ball travels back up. The downward bounces form a geometric series: